\(\int \frac {x^2 \sqrt {a+b x}}{(c+d x)^{5/2}} \, dx\) [208]

Optimal result

Integrand size = 22, antiderivative size = 136 \[ \int \frac {x^2 \sqrt {a+b x}}{(c+d x)^{5/2}} \, dx=\frac {2 c^2 (a+b x)^{3/2}}{3 d^2 (b c-a d) (c+d x)^{3/2}}+\frac {4 c \sqrt {a+b x}}{d^3 \sqrt {c+d x}}+\frac {\sqrt {a+b x} \sqrt {c+d x}}{d^3}-\frac {(5 b c-a d) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{\sqrt {b} d^{7/2}} \] Output:

2/3*c^2*(b*x+a)^(3/2)/d^2/(-a*d+b*c)/(d*x+c)^(3/2)+4*c*(b*x+a)^(1/2)/d^3/( 
d*x+c)^(1/2)+(b*x+a)^(1/2)*(d*x+c)^(1/2)/d^3-(-a*d+5*b*c)*arctanh(d^(1/2)* 
(b*x+a)^(1/2)/b^(1/2)/(d*x+c)^(1/2))/b^(1/2)/d^(7/2)
 

Mathematica [A] (verified)

Time = 0.22 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.96 \[ \int \frac {x^2 \sqrt {a+b x}}{(c+d x)^{5/2}} \, dx=\frac {\sqrt {a+b x} \left (a d \left (13 c^2+18 c d x+3 d^2 x^2\right )-b c \left (15 c^2+20 c d x+3 d^2 x^2\right )\right )}{3 d^3 (-b c+a d) (c+d x)^{3/2}}+\frac {(-5 b c+a d) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{\sqrt {b} d^{7/2}} \] Input:

Integrate[(x^2*Sqrt[a + b*x])/(c + d*x)^(5/2),x]
 

Output:

(Sqrt[a + b*x]*(a*d*(13*c^2 + 18*c*d*x + 3*d^2*x^2) - b*c*(15*c^2 + 20*c*d 
*x + 3*d^2*x^2)))/(3*d^3*(-(b*c) + a*d)*(c + d*x)^(3/2)) + ((-5*b*c + a*d) 
*ArcTanh[(Sqrt[d]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[c + d*x])])/(Sqrt[b]*d^(7/2 
))
 

Rubi [A] (verified)

Time = 0.25 (sec) , antiderivative size = 170, normalized size of antiderivative = 1.25, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {100, 27, 87, 60, 66, 221}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {x^2 \sqrt {a+b x}}{(c+d x)^{5/2}} \, dx\)

\(\Big \downarrow \) 100

\(\displaystyle \frac {2 c^2 (a+b x)^{3/2}}{3 d^2 (c+d x)^{3/2} (b c-a d)}-\frac {2 \int \frac {3 (b c-a d) \sqrt {a+b x} (c-d x)}{2 (c+d x)^{3/2}}dx}{3 d^2 (b c-a d)}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {2 c^2 (a+b x)^{3/2}}{3 d^2 (c+d x)^{3/2} (b c-a d)}-\frac {\int \frac {\sqrt {a+b x} (c-d x)}{(c+d x)^{3/2}}dx}{d^2}\)

\(\Big \downarrow \) 87

\(\displaystyle \frac {2 c^2 (a+b x)^{3/2}}{3 d^2 (c+d x)^{3/2} (b c-a d)}-\frac {\frac {4 c (a+b x)^{3/2}}{\sqrt {c+d x} (b c-a d)}-\frac {(5 b c-a d) \int \frac {\sqrt {a+b x}}{\sqrt {c+d x}}dx}{b c-a d}}{d^2}\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {2 c^2 (a+b x)^{3/2}}{3 d^2 (c+d x)^{3/2} (b c-a d)}-\frac {\frac {4 c (a+b x)^{3/2}}{\sqrt {c+d x} (b c-a d)}-\frac {(5 b c-a d) \left (\frac {\sqrt {a+b x} \sqrt {c+d x}}{d}-\frac {(b c-a d) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}}dx}{2 d}\right )}{b c-a d}}{d^2}\)

\(\Big \downarrow \) 66

\(\displaystyle \frac {2 c^2 (a+b x)^{3/2}}{3 d^2 (c+d x)^{3/2} (b c-a d)}-\frac {\frac {4 c (a+b x)^{3/2}}{\sqrt {c+d x} (b c-a d)}-\frac {(5 b c-a d) \left (\frac {\sqrt {a+b x} \sqrt {c+d x}}{d}-\frac {(b c-a d) \int \frac {1}{b-\frac {d (a+b x)}{c+d x}}d\frac {\sqrt {a+b x}}{\sqrt {c+d x}}}{d}\right )}{b c-a d}}{d^2}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {2 c^2 (a+b x)^{3/2}}{3 d^2 (c+d x)^{3/2} (b c-a d)}-\frac {\frac {4 c (a+b x)^{3/2}}{\sqrt {c+d x} (b c-a d)}-\frac {(5 b c-a d) \left (\frac {\sqrt {a+b x} \sqrt {c+d x}}{d}-\frac {(b c-a d) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{\sqrt {b} d^{3/2}}\right )}{b c-a d}}{d^2}\)

Input:

Int[(x^2*Sqrt[a + b*x])/(c + d*x)^(5/2),x]
 

Output:

(2*c^2*(a + b*x)^(3/2))/(3*d^2*(b*c - a*d)*(c + d*x)^(3/2)) - ((4*c*(a + b 
*x)^(3/2))/((b*c - a*d)*Sqrt[c + d*x]) - ((5*b*c - a*d)*((Sqrt[a + b*x]*Sq 
rt[c + d*x])/d - ((b*c - a*d)*ArcTanh[(Sqrt[d]*Sqrt[a + b*x])/(Sqrt[b]*Sqr 
t[c + d*x])])/(Sqrt[b]*d^(3/2))))/(b*c - a*d))/d^2
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 60

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ 
(a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + n + 1))), x] + Simp[n*((b*c - a*d)/( 
b*(m + n + 1)))   Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, 
 c, d}, x] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !Integer 
Q[n] || (GtQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinear 
Q[a, b, c, d, m, n, x]
 

rule 66

Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[ 
2   Subst[Int[1/(b - d*x^2), x], x, Sqrt[a + b*x]/Sqrt[c + d*x]], x] /; Fre 
eQ[{a, b, c, d}, x] &&  !GtQ[c - a*(d/b), 0]
 

rule 87

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p 
_.), x_] :> Simp[(-(b*e - a*f))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p 
+ 1)*(c*f - d*e))), x] - Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p 
+ 1)))/(f*(p + 1)*(c*f - d*e))   Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] 
/; FreeQ[{a, b, c, d, e, f, n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || Intege 
rQ[p] ||  !(IntegerQ[n] ||  !(EqQ[e, 0] ||  !(EqQ[c, 0] || LtQ[p, n]))))
 

rule 100

Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^( 
p_), x_] :> Simp[(b*c - a*d)^2*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d^2*(d 
*e - c*f)*(n + 1))), x] - Simp[1/(d^2*(d*e - c*f)*(n + 1))   Int[(c + d*x)^ 
(n + 1)*(e + f*x)^p*Simp[a^2*d^2*f*(n + p + 2) + b^2*c*(d*e*(n + 1) + c*f*( 
p + 1)) - 2*a*b*d*(d*e*(n + 1) + c*f*(p + 1)) - b^2*d*(d*e - c*f)*(n + 1)*x 
, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && (LtQ[n, -1] || (EqQ[n 
 + p + 3, 0] && NeQ[n, -1] && (SumSimplerQ[n, 1] ||  !SumSimplerQ[p, 1])))
 

rule 221

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x 
/Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(658\) vs. \(2(110)=220\).

Time = 0.22 (sec) , antiderivative size = 659, normalized size of antiderivative = 4.85

method	result	size
default	\(\frac {\left (3 \ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {d b}+a d +b c}{2 \sqrt {d b}}\right ) a^{2} d^{4} x^{2}-18 \ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {d b}+a d +b c}{2 \sqrt {d b}}\right ) a b c \,d^{3} x^{2}+15 \ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {d b}+a d +b c}{2 \sqrt {d b}}\right ) b^{2} c^{2} d^{2} x^{2}+6 \ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {d b}+a d +b c}{2 \sqrt {d b}}\right ) a^{2} c \,d^{3} x -36 \ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {d b}+a d +b c}{2 \sqrt {d b}}\right ) a b \,c^{2} d^{2} x +30 \ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {d b}+a d +b c}{2 \sqrt {d b}}\right ) b^{2} c^{3} d x +6 \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {d b}\, a \,d^{3} x^{2}-6 \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {d b}\, b c \,d^{2} x^{2}+3 \ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {d b}+a d +b c}{2 \sqrt {d b}}\right ) a^{2} c^{2} d^{2}-18 \ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {d b}+a d +b c}{2 \sqrt {d b}}\right ) a b \,c^{3} d +15 \ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {d b}+a d +b c}{2 \sqrt {d b}}\right ) b^{2} c^{4}+36 \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {d b}\, a c \,d^{2} x -40 \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {d b}\, b \,c^{2} d x +26 \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {d b}\, a \,c^{2} d -30 \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {d b}\, b \,c^{3}\right ) \sqrt {b x +a}}{6 \left (a d -b c \right ) \sqrt {d b}\, \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, d^{3} \left (x d +c \right )^{\frac {3}{2}}}\)	\(659\)

Input:

int(x^2*(b*x+a)^(1/2)/(d*x+c)^(5/2),x,method=_RETURNVERBOSE)
 

Output:

1/6*(3*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(d*b)^(1/2)+a*d+b*c)/(d*b 
)^(1/2))*a^2*d^4*x^2-18*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(d*b)^(1 
/2)+a*d+b*c)/(d*b)^(1/2))*a*b*c*d^3*x^2+15*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x 
+c))^(1/2)*(d*b)^(1/2)+a*d+b*c)/(d*b)^(1/2))*b^2*c^2*d^2*x^2+6*ln(1/2*(2*b 
*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(d*b)^(1/2)+a*d+b*c)/(d*b)^(1/2))*a^2*c*d^3 
*x-36*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(d*b)^(1/2)+a*d+b*c)/(d*b) 
^(1/2))*a*b*c^2*d^2*x+30*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(d*b)^( 
1/2)+a*d+b*c)/(d*b)^(1/2))*b^2*c^3*d*x+6*((b*x+a)*(d*x+c))^(1/2)*(d*b)^(1/ 
2)*a*d^3*x^2-6*((b*x+a)*(d*x+c))^(1/2)*(d*b)^(1/2)*b*c*d^2*x^2+3*ln(1/2*(2 
*b*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(d*b)^(1/2)+a*d+b*c)/(d*b)^(1/2))*a^2*c^2 
*d^2-18*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(d*b)^(1/2)+a*d+b*c)/(d* 
b)^(1/2))*a*b*c^3*d+15*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(d*b)^(1/ 
2)+a*d+b*c)/(d*b)^(1/2))*b^2*c^4+36*((b*x+a)*(d*x+c))^(1/2)*(d*b)^(1/2)*a* 
c*d^2*x-40*((b*x+a)*(d*x+c))^(1/2)*(d*b)^(1/2)*b*c^2*d*x+26*((b*x+a)*(d*x+ 
c))^(1/2)*(d*b)^(1/2)*a*c^2*d-30*((b*x+a)*(d*x+c))^(1/2)*(d*b)^(1/2)*b*c^3 
)*(b*x+a)^(1/2)/(a*d-b*c)/(d*b)^(1/2)/((b*x+a)*(d*x+c))^(1/2)/d^3/(d*x+c)^ 
(3/2)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 313 vs. \(2 (110) = 220\).

Time = 0.21 (sec) , antiderivative size = 640, normalized size of antiderivative = 4.71 \[ \int \frac {x^2 \sqrt {a+b x}}{(c+d x)^{5/2}} \, dx=\left [-\frac {3 \, {\left (5 \, b^{2} c^{4} - 6 \, a b c^{3} d + a^{2} c^{2} d^{2} + {\left (5 \, b^{2} c^{2} d^{2} - 6 \, a b c d^{3} + a^{2} d^{4}\right )} x^{2} + 2 \, {\left (5 \, b^{2} c^{3} d - 6 \, a b c^{2} d^{2} + a^{2} c d^{3}\right )} x\right )} \sqrt {b d} \log \left (8 \, b^{2} d^{2} x^{2} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} + 4 \, {\left (2 \, b d x + b c + a d\right )} \sqrt {b d} \sqrt {b x + a} \sqrt {d x + c} + 8 \, {\left (b^{2} c d + a b d^{2}\right )} x\right ) - 4 \, {\left (15 \, b^{2} c^{3} d - 13 \, a b c^{2} d^{2} + 3 \, {\left (b^{2} c d^{3} - a b d^{4}\right )} x^{2} + 2 \, {\left (10 \, b^{2} c^{2} d^{2} - 9 \, a b c d^{3}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{12 \, {\left (b^{2} c^{3} d^{4} - a b c^{2} d^{5} + {\left (b^{2} c d^{6} - a b d^{7}\right )} x^{2} + 2 \, {\left (b^{2} c^{2} d^{5} - a b c d^{6}\right )} x\right )}}, \frac {3 \, {\left (5 \, b^{2} c^{4} - 6 \, a b c^{3} d + a^{2} c^{2} d^{2} + {\left (5 \, b^{2} c^{2} d^{2} - 6 \, a b c d^{3} + a^{2} d^{4}\right )} x^{2} + 2 \, {\left (5 \, b^{2} c^{3} d - 6 \, a b c^{2} d^{2} + a^{2} c d^{3}\right )} x\right )} \sqrt {-b d} \arctan \left (\frac {{\left (2 \, b d x + b c + a d\right )} \sqrt {-b d} \sqrt {b x + a} \sqrt {d x + c}}{2 \, {\left (b^{2} d^{2} x^{2} + a b c d + {\left (b^{2} c d + a b d^{2}\right )} x\right )}}\right ) + 2 \, {\left (15 \, b^{2} c^{3} d - 13 \, a b c^{2} d^{2} + 3 \, {\left (b^{2} c d^{3} - a b d^{4}\right )} x^{2} + 2 \, {\left (10 \, b^{2} c^{2} d^{2} - 9 \, a b c d^{3}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{6 \, {\left (b^{2} c^{3} d^{4} - a b c^{2} d^{5} + {\left (b^{2} c d^{6} - a b d^{7}\right )} x^{2} + 2 \, {\left (b^{2} c^{2} d^{5} - a b c d^{6}\right )} x\right )}}\right ] \] Input:

integrate(x^2*(b*x+a)^(1/2)/(d*x+c)^(5/2),x, algorithm="fricas")
 

Output:

[-1/12*(3*(5*b^2*c^4 - 6*a*b*c^3*d + a^2*c^2*d^2 + (5*b^2*c^2*d^2 - 6*a*b* 
c*d^3 + a^2*d^4)*x^2 + 2*(5*b^2*c^3*d - 6*a*b*c^2*d^2 + a^2*c*d^3)*x)*sqrt 
(b*d)*log(8*b^2*d^2*x^2 + b^2*c^2 + 6*a*b*c*d + a^2*d^2 + 4*(2*b*d*x + b*c 
 + a*d)*sqrt(b*d)*sqrt(b*x + a)*sqrt(d*x + c) + 8*(b^2*c*d + a*b*d^2)*x) - 
 4*(15*b^2*c^3*d - 13*a*b*c^2*d^2 + 3*(b^2*c*d^3 - a*b*d^4)*x^2 + 2*(10*b^ 
2*c^2*d^2 - 9*a*b*c*d^3)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(b^2*c^3*d^4 - a* 
b*c^2*d^5 + (b^2*c*d^6 - a*b*d^7)*x^2 + 2*(b^2*c^2*d^5 - a*b*c*d^6)*x), 1/ 
6*(3*(5*b^2*c^4 - 6*a*b*c^3*d + a^2*c^2*d^2 + (5*b^2*c^2*d^2 - 6*a*b*c*d^3 
 + a^2*d^4)*x^2 + 2*(5*b^2*c^3*d - 6*a*b*c^2*d^2 + a^2*c*d^3)*x)*sqrt(-b*d 
)*arctan(1/2*(2*b*d*x + b*c + a*d)*sqrt(-b*d)*sqrt(b*x + a)*sqrt(d*x + c)/ 
(b^2*d^2*x^2 + a*b*c*d + (b^2*c*d + a*b*d^2)*x)) + 2*(15*b^2*c^3*d - 13*a* 
b*c^2*d^2 + 3*(b^2*c*d^3 - a*b*d^4)*x^2 + 2*(10*b^2*c^2*d^2 - 9*a*b*c*d^3) 
*x)*sqrt(b*x + a)*sqrt(d*x + c))/(b^2*c^3*d^4 - a*b*c^2*d^5 + (b^2*c*d^6 - 
 a*b*d^7)*x^2 + 2*(b^2*c^2*d^5 - a*b*c*d^6)*x)]
 

Sympy [F]

\[ \int \frac {x^2 \sqrt {a+b x}}{(c+d x)^{5/2}} \, dx=\int \frac {x^{2} \sqrt {a + b x}}{\left (c + d x\right )^{\frac {5}{2}}}\, dx \] Input:

integrate(x**2*(b*x+a)**(1/2)/(d*x+c)**(5/2),x)
 

Output:

Integral(x**2*sqrt(a + b*x)/(c + d*x)**(5/2), x)
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {x^2 \sqrt {a+b x}}{(c+d x)^{5/2}} \, dx=\text {Exception raised: ValueError} \] Input:

integrate(x^2*(b*x+a)^(1/2)/(d*x+c)^(5/2),x, algorithm="maxima")
 

Output:

Exception raised: ValueError >> Computation failed since Maxima requested 
additional constraints; using the 'assume' command before evaluation *may* 
 help (example of legal syntax is 'assume(a*d-b*c>0)', see `assume?` for m 
ore detail
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 283 vs. \(2 (110) = 220\).

Time = 0.19 (sec) , antiderivative size = 283, normalized size of antiderivative = 2.08 \[ \int \frac {x^2 \sqrt {a+b x}}{(c+d x)^{5/2}} \, dx=\frac {{\left ({\left (b x + a\right )} {\left (\frac {3 \, {\left (b^{6} c d^{4} - a b^{5} d^{5}\right )} {\left (b x + a\right )}}{b^{4} c d^{5} {\left | b \right |} - a b^{3} d^{6} {\left | b \right |}} + \frac {2 \, {\left (10 \, b^{7} c^{2} d^{3} - 12 \, a b^{6} c d^{4} + 3 \, a^{2} b^{5} d^{5}\right )}}{b^{4} c d^{5} {\left | b \right |} - a b^{3} d^{6} {\left | b \right |}}\right )} + \frac {3 \, {\left (5 \, b^{8} c^{3} d^{2} - 11 \, a b^{7} c^{2} d^{3} + 7 \, a^{2} b^{6} c d^{4} - a^{3} b^{5} d^{5}\right )}}{b^{4} c d^{5} {\left | b \right |} - a b^{3} d^{6} {\left | b \right |}}\right )} \sqrt {b x + a}}{3 \, {\left (b^{2} c + {\left (b x + a\right )} b d - a b d\right )}^{\frac {3}{2}}} + \frac {{\left (5 \, b^{2} c - a b d\right )} \log \left ({\left | -\sqrt {b d} \sqrt {b x + a} + \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d} \right |}\right )}{\sqrt {b d} d^{3} {\left | b \right |}} \] Input:

integrate(x^2*(b*x+a)^(1/2)/(d*x+c)^(5/2),x, algorithm="giac")
 

Output:

1/3*((b*x + a)*(3*(b^6*c*d^4 - a*b^5*d^5)*(b*x + a)/(b^4*c*d^5*abs(b) - a* 
b^3*d^6*abs(b)) + 2*(10*b^7*c^2*d^3 - 12*a*b^6*c*d^4 + 3*a^2*b^5*d^5)/(b^4 
*c*d^5*abs(b) - a*b^3*d^6*abs(b))) + 3*(5*b^8*c^3*d^2 - 11*a*b^7*c^2*d^3 + 
 7*a^2*b^6*c*d^4 - a^3*b^5*d^5)/(b^4*c*d^5*abs(b) - a*b^3*d^6*abs(b)))*sqr 
t(b*x + a)/(b^2*c + (b*x + a)*b*d - a*b*d)^(3/2) + (5*b^2*c - a*b*d)*log(a 
bs(-sqrt(b*d)*sqrt(b*x + a) + sqrt(b^2*c + (b*x + a)*b*d - a*b*d)))/(sqrt( 
b*d)*d^3*abs(b))
 

Mupad [F(-1)]

Timed out. \[ \int \frac {x^2 \sqrt {a+b x}}{(c+d x)^{5/2}} \, dx=\int \frac {x^2\,\sqrt {a+b\,x}}{{\left (c+d\,x\right )}^{5/2}} \,d x \] Input:

int((x^2*(a + b*x)^(1/2))/(c + d*x)^(5/2),x)
 

Output:

int((x^2*(a + b*x)^(1/2))/(c + d*x)^(5/2), x)
 

Reduce [B] (verification not implemented)

Time = 0.26 (sec) , antiderivative size = 738, normalized size of antiderivative = 5.43 \[ \int \frac {x^2 \sqrt {a+b x}}{(c+d x)^{5/2}} \, dx =\text {Too large to display} \] Input:

int(x^2*(b*x+a)^(1/2)/(d*x+c)^(5/2),x)
 

Output:

(26*sqrt(c + d*x)*sqrt(a + b*x)*a*b*c**2*d**2 + 36*sqrt(c + d*x)*sqrt(a + 
b*x)*a*b*c*d**3*x + 6*sqrt(c + d*x)*sqrt(a + b*x)*a*b*d**4*x**2 - 30*sqrt( 
c + d*x)*sqrt(a + b*x)*b**2*c**3*d - 40*sqrt(c + d*x)*sqrt(a + b*x)*b**2*c 
**2*d**2*x - 6*sqrt(c + d*x)*sqrt(a + b*x)*b**2*c*d**3*x**2 + 6*sqrt(d)*sq 
rt(b)*log((sqrt(d)*sqrt(a + b*x) + sqrt(b)*sqrt(c + d*x))/sqrt(a*d - b*c)) 
*a**2*c**2*d**2 + 12*sqrt(d)*sqrt(b)*log((sqrt(d)*sqrt(a + b*x) + sqrt(b)* 
sqrt(c + d*x))/sqrt(a*d - b*c))*a**2*c*d**3*x + 6*sqrt(d)*sqrt(b)*log((sqr 
t(d)*sqrt(a + b*x) + sqrt(b)*sqrt(c + d*x))/sqrt(a*d - b*c))*a**2*d**4*x** 
2 - 36*sqrt(d)*sqrt(b)*log((sqrt(d)*sqrt(a + b*x) + sqrt(b)*sqrt(c + d*x)) 
/sqrt(a*d - b*c))*a*b*c**3*d - 72*sqrt(d)*sqrt(b)*log((sqrt(d)*sqrt(a + b* 
x) + sqrt(b)*sqrt(c + d*x))/sqrt(a*d - b*c))*a*b*c**2*d**2*x - 36*sqrt(d)* 
sqrt(b)*log((sqrt(d)*sqrt(a + b*x) + sqrt(b)*sqrt(c + d*x))/sqrt(a*d - b*c 
))*a*b*c*d**3*x**2 + 30*sqrt(d)*sqrt(b)*log((sqrt(d)*sqrt(a + b*x) + sqrt( 
b)*sqrt(c + d*x))/sqrt(a*d - b*c))*b**2*c**4 + 60*sqrt(d)*sqrt(b)*log((sqr 
t(d)*sqrt(a + b*x) + sqrt(b)*sqrt(c + d*x))/sqrt(a*d - b*c))*b**2*c**3*d*x 
 + 30*sqrt(d)*sqrt(b)*log((sqrt(d)*sqrt(a + b*x) + sqrt(b)*sqrt(c + d*x))/ 
sqrt(a*d - b*c))*b**2*c**2*d**2*x**2 + sqrt(d)*sqrt(b)*a**2*c**2*d**2 + 2* 
sqrt(d)*sqrt(b)*a**2*c*d**3*x + sqrt(d)*sqrt(b)*a**2*d**4*x**2 - 10*sqrt(d 
)*sqrt(b)*a*b*c**3*d - 20*sqrt(d)*sqrt(b)*a*b*c**2*d**2*x - 10*sqrt(d)*sqr 
t(b)*a*b*c*d**3*x**2 + 5*sqrt(d)*sqrt(b)*b**2*c**4 + 10*sqrt(d)*sqrt(b)...